Below is an extract from the book "The Talmud: a Biography" by Jewish author Harry Freedman. I'll refrain from the usual practice of putting quotes in italics since it can be weary on the eye with such a large amount of text. But everything below is quoted.

In the year 622 ce the prophet Mohammed and his followers embarked on a series of military campaigns from their base at Medina in the Arabian Peninsula. Within a remarkably short period of time the political and religious map of the Middle East would look very different. No nation, faith or institution which fell under their influence would emerge unchanged. The Talmud was no exception.

Mohammed had an extensive knowledge of Judaism and Jewish practice. This is clear from the Qu’ran itself, as well as from later commentaries and legends. He may even have been influenced by Jewish teachers. One legend has the Prophet discussing the names of the stars in Joseph’s dream with the son of the exilarch Bustanai, of whom we will soon hear more. At first Mohammed instructed his followers to face Jerusalem when praying, as the Jews do, and he instituted a fast on the tenth day of the first month, corresponding to the Jewish Day of Atonement.

When he eventually abandoned these practices it was most likely the result of an early alliance with local Jewish tribes turning sour.

Medina, in modern Saudi Arabia, is several hundred miles south-east of the Land of Israel. Jewish tribes were amongst its earliest inhabitants; Moshe Gil explains that the first Jewish settlers in the Arabian Peninsula were refugees from the Romans whose numbers increased as they converted the surrounding Arabian tribes.

Unlike the nomadic Bedouins, with whom they shared the region, the Jews lived in walled towns and farmed the land, growing dates and vines. The Jewish tribes played an active role in the governance of Medina. In fact their presence in the Arabian Peninsula was so influential that for a short period in the sixth century the royal household of Yemen converted to Judaism.

Early relations between Mohammed’s followers and the Jews in the area were good. In the Constitution of Medina, which Muslim sources describe as a pact between the Muslims and the Jews, Mohammed states that ‘the Jews have their religion and the Muslims have theirs’.

But the amity wouldn’t last. Despite the contact they had with the Prophet, not all of the Jews of Medina warmed to the Islamic revolution. Mohammed fought and won separate battles against three different Jewish tribes, expelling two of them, massacring most of the men in the third and taking the women and children into slavery.

It has been argued that the hostility of the Arabian Jewish tribes to Mohammed was due to his alliances with dissenting Jewish sects. Our modern conception of religion prevents us from appreciating just how fluid the ancient faiths were. Major religions today have a clear set of doctrines, well-established rituals, clergy who are responsible for the faith’s propagation and buildings dedicated to worship. But the early days of a faith are rarely so well structured. The history of Judaism, Christianity and Islam is full of competing sects vying for influence.

The Talmud sat at the centre of Jewish life in Babylon and Israel but it was probably unknown to the remote desert communities. Even if the Jews of Arabia were aware of it, there is no reason to assume that its teachings remained unchanged as they diffused from the centre to the periphery of Jewish settlement.

Far from it; the Talmudic centres were so distant from Arabia, and the prevailing lifestyles so different that we can imagine a local, exotic Jewish culture which would have been virtually unrecognizable to a Jew from Babylon.
The Mishnah had cemented the religious authority of the rabbis amongst the Jewish communities in Israel and Babylon, but that wasn’t the case elsewhere. The ideological victory that the Pharisees had gained over the Sadducees all those years ago hadn’t universally standardized Jewish belief and practice. Nor had the birth of Christianity stopped other messianic groups from emerging within Judaism; the best known of whom were followers of the eighth–century, sectarian Abu Isa.

One of the ideological battles the Talmud was yet to fight would be to bring dissenting Jewish groups, such as those in the Arabian Peninsula, within its sphere of influence. It was the spread of Islam over the next few centuries which allowed the battle to be won.

The highly charged intellectual environment in Baghdad created one of the two conditions the Talmud needed if it was to break out of the academies into the wider world. The other was Baghdad’s geographical location, at the centre of the caliphate’s unified polity and international communication network, stretching from the Atlantic coast of North Africa to the borders of India.

More than anything else it was these two factors which would bring about and facilitate the dissemination of the Babylonian Talmud to Jewish communities throughout the Islamic world, and eventually beyond.
As their links with the new metropolis strengthened the luminaries of Sura and Pumbedita found they had much in common with their opposite numbers in the Islamic world. They discovered they were grappling with similar issues, actively applying their scholarship, legal and religious traditions, to regulate the day-to-day lives of their co-religionists.

Each faith influenced the other. This is obvious both from the structure of their legal systems and some of the legislation itself. Their influence upon each other was more than just a simple two-way process; Gideon Libson explains it as a feedback model in which the Talmudic system first impacted Islam, which at a later stage left its imprint on Talmudic law.

Both Islam and Judaism are religions which minutely regulate every aspect of the believer’s life. They’re each based on a God-given written document – the Torah for Judaism and the Qu’ran for Islam. These divine texts are each interpreted and expanded upon by an oral tradition – the Talmud and the Hadith respectively. Both traditions contain legal and ethical material, and the legal material in each distinguishes between religious laws and social laws. The Jewish system of law is called halacha, the Islamic system is called shar’ia. Both names mean a ‘pathway’ or a ‘way to go’.

Unlike Christianity, the laws and beliefs in Islam and Judaism are derived through a process of reasoning and scholarship; there are no councils or synods to rule on doctrine, ethics or behaviour. In fact the two religions are so close in terms of their structure that the tenth-century rabbinic leader Saadia Gaon would unselfconsciously refer to Jewish law as shar’ia, to the prayer leader in a synagogue as an imam and the direction in which Jews faced when praying as qibla.

Although in these twin systems the Qu’ran and Torah parallel each other as divinely revealed written texts, the Qu’ran was written down long after the Torah. So whilst we find characters, ideas and motifs from the Torah in the Qu’ran, we shouldn’t expect to find them the other way round.

However, the Talmud and Qu’ran do originate from a similar period and it’s not hard to find concepts from the Talmud occurring in both the Qu’ran and Hadith, and vice versa.
One such case is the ritual definition of daybreak. The Qu’ran defines the moment of daybreak, when the faithful must begin to fast during Ramadan, as when the ‘white thread of dawn becomes distinct to you from the black thread’. Similarly, the Jewish Mishnah rules that the morning prayer is to be said when the worshipper can distinguish between blue and white.

Both traditions use the same analogy to emphasize the sanctity and uniqueness of every human life. The Qu’ran states that ‘We decreed upon the Children of Israel that whoever kills a soul … it is as if he had slain mankind entirely. And whoever saves one – it is as if he had saved mankind entirely’. This is a reference to the passage from the Mishnah that:
Adam was created alone to teach you that whoever takes a human life is considered by the Bible to have destroyed an entire world. And whoever saves a human life is considered by the Bible as if he preserved an entire world.

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An extensive survey by Gideon Libson has shown similarities between the rulings of a tenth-century gaon, Shmuel ben Hofni, and Islamic legal writing of the same period. Of course, as Libson concedes, the fact that there are similarities between two legal rulings in different systems doesn’t necessarily mean that one system was dependent upon the other; they may both have independently derived their rulings from a third source that they each knew.

But with all the other evidence of contact and cross-influences between the faiths, it’s pretty likely that the Talmudic and Islamic systems of law influenced each other.
Talmudic and Islamic scholars cross-fertilized in legal matters because they lived in the same mercantile society. But the two traditions didn’t just overlap when it came to the law.

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Both the Talmudic and Islamic scholars shared the same imperative, to extend their authority beyond their immediate neighbourhood. It was important for the Islamic scholars because the faith was relatively new, the empire was still expanding and there were many areas that were not yet fully Islamized. It mattered to the Talmudic scholars because the small Jewish nation was scattered over a vast geographical area, and the danger was that without a centre of influence the faith may weaken and dissipate.

There was also the question of the Jerusalem Talmud, which still exerted its authority in regions which the Babylonians wished to reach. Although Israel was the ancestral homeland its impoverished community was lacking in leadership and serious legal scholarship. Its main cultural activity had for some time been the production of poetry and biblical interpretation; disciplines which the scholarly geonim considered relatively unimportant. In the early days Babylon had accepted the authority of the scholars in the Holy Land without question; now this was no longer the case. The last time we hear of a Babylonian scholar applying to the Academy in Israel for the answer to a problem was in the late fifth century.

As scholarship in one centre blossomed and the other faded the contest between the two Talmuds for influence was to grow increasingly trenchant. The Babylonian Talmud’s position at the heart of the Islamic Empire gave its advocates an advantage over its hard-pressed rivals in Israel; the majority of Jews lived under Islamic rule and it was far easier to communicate with them from Baghdad than from Jerusalem.

Additionally, and perhaps most importantly, the sectarian threat had not been dispelled. There was a growing body of dissenting voices within the Jewish world; sects which had never been part of the rabbinic camp or which had become disillusioned with it and now, tolerated by the caliphate, were becoming more vociferous. Spreading the Talmud far and wide was one of the strategies the geonim would try to use to bring these communities into the mainstream.

It was no coincidence that the scholars of both faiths used identical tools to promote the fruits of their labours. One was the new science of codification; which reduced complex legal discussions to systematic, logically arranged summations of the law. The other was the well-established craft of letter writing.

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Codification enabled people to access a concise summary of the law but codes in themselves didn’t help much in building relationships between the academies and distant communities. A far more effective way for the Talmudic and Sharia scholars to establish their authority was by corresponding with their co-religionists, answering their questions and providing religious guidance. The Jews called these answers teshuvot; the Muslims called them fatwas. Academics of all faiths and none call them responsa.